New Scientist is reporting that scientists have created the fastest spinning object ever: a fleck of graphene spun up to an incredible rate of a million rotations per second!

Normally, while very cool, that’s not the sort of thing I’d write about here. But I had an idle moment, and wondered about what that rotation really meant. I did a little math, and came up with some astonishing numbers.

First off, graphene is a flat sheet made up of carbon atoms; each atom connects to the others in a hexagon pattern, and the sheet is only one atom thick! This sheet is incredibly strong, and scientists are excited by it because if it can be produced on a large scale it would have tremendous use.

In this case, the scientists created tiny flakes of it only a micron (one-millionth of a meter) across; that’s about 1/50th the width of a human hair. They suspended the flakes in a chamber using electric fields, then spun them up using a beam of light.

I started picturing what that must be like, these tiny whirling motes of carbon, and realized that the forces on a spinning flake must be huge. And by huge, I mean monstrous. When you spin, you feel a force called the centripetal force. It’s what you feel when you’re on a merry-go-round, or a car making a turn (it’s the same thing as centrifugal force, just seen in a different way). The magnitude of this force, how strong it is, depends on how fast you’re moving, and how big a circle you’re making.

I decided to calculate the size of the force on a flake. As it spins, the corners are making the biggest circles, so they have the highest centripetal force on them. To make things easy, let’s assume the distance from the center of the flake to a corner is one micron. We need the velocity of the corner as it spins, which is the distance traveled divided by the time. Well, it makes a complete circle once per one-millionth of a second. And the distance is the circumference of the circle it makes, which is 2 x π x radius. That means:

That’s only about 20 feet/second, or roughly 15 mph. I can bike faster than that.

However, remember, we’re talking about an object smaller than a human hair! So that’s a tremendous speed. The force on it is incredible. Centripetal acceleration* is

velocity2 / radius

or

(6.3 meters/second)2 / 1 micron = 4 x 107 meter/second2

If that sounds like a lot, it is: the acceleration of gravity on the Earth’s surface is about 10 meter/second2, so the outwards force you’d feel on that flake would be about four million times the Earth’s gravity.

Just to give you an idea of how much force that is, I weigh about 170 pounds. If I were under that kind of acceleration, it would feel like hanging from a rope… with a fully loaded oil tanker tied to my feet.

Ow.

Almost anything on Earth would fly apart at that kind of acceleration, but graphene is just that strong. In fact, the scientists involved said it could withstand even higher speeds! So you can probably get an idea of why this stuff is so cool, and why it might have some interesting applications.

I’m thinking roller coasters. You wouldn’t even need seat belts to keep people in their seats… but you’d need spatulas to get them off.

* I switched to acceleration here because it makes the math and the explanation a bit easier. In the end I’ve been careful about terminology –acceleration and force are related but different — and I apologize for any confusion.

Even some high-mass main sequence stars spin astoundingly fast. So much so that some – three notable examples being Altair, Regulus and, most of all, Achernar – are distorted from spherical globes to elliptical egg-shapes.

If there were microsecond pulsars, that might be close. A millisecond pulsar is more than 1000 times slower. Still bloody fast, though….

Diamond is *harder* than graphene. That’s not the same as saying it’s stronger. Diamond isn’t that strong; you can crush it with a common hammer, far less than a million G’s. The key here appears to be applying the forces gradually; if you somehow attached a cable to each corner and yanked with a million times its weight, I suspect the graphene would break. But the centripetal force is applied continuously across the medium, gradually shrinking to zero at the center of rotation.

It confuses beginning physicists, but the real centripetal acceleration formula is (rotation rate)^2 * radius. The rotation rate is “a million times per second.” This is mathematically equivalent to Phil’s formula, but it doesn’t look singular in the middle (Phil’s isn’t either; it’s just not as obvious).

Somebody tell me I’m crazy, but when I read about tremendous accelerations like this, I can’t help but wonder what the relativistic effects on such a system be…

I mean, general relativity says that acceleration is the same as gravity, at least to an internal observer, so does that apply here, or is there something that I don’t understand/know about that counteracts the affects of relativity, specifically time dilation and that sort of thing. More specifically, what would it feel like to be riding on a carbon atom on the edge of this thing? (While holding on for dear life!) Is this sort of thing bending space and slowing down time and things of that nature?

Also, how does 4×10^6 g compare to the gravity of very massive compact objects, like say black holes?

@4. Tree Lobsters : Me too. Space elevators would be the ideal solution to the problem of getting material (& people) out of Earth’s gravity well and into orbit if we could actually build them, methinks.

Actually you can make graphene at home right now. Take a pencil and piece of paper. Make a line. OK, you did it! There are bits of graphene on the paper. The first way they found graphene was by taking a piece of scotch tape to a piece of graphite and pulling off the layers. Unfortunately the pieces you get are small and it’s not very economical right now, but they hope to be able to mass produce larger sheets soon. You might see it in thin displays in a few years.

One may wonder what (if any) centripetal forces are acting on our Universe.
Planets spin, Back holes spin, why not the entire Universe. Maybe this accounts for some dark energy forcing us apart and increasing expansion. Hard to tell as there is no frame of reference.

Limiting equatorial velocity for a solid sphere depends only on the material’s binding energy and is independent of radius. The interested reader is invited to derive the following for a circular sheet.

Limiting equatorial velocity for a rotating solid sphere (independent of the sphere’s radius!) is given by

v_lim = sqrt[(2)(S)/(rho)]

where S is the yield strength and rho the density. Diamond tensile strength does not exceed 225 GPa with density 3520 kg/m^3. Convert Pa to kg/m^2 at one gee.

Unbinding a body by spin is straighforeward. S is yield strength, rho is density, based on forces acting upon a surface element with area A of the rotating body. For an infinitesimally thin surface element, stresses are tangential as radial stresses go to zero on the surface. Said tangential stresses still yield a radial force component for a curved surface. From the theory of elasticity: given a surface element with area dA and thickness dt, tangential stresses present in the surface, the normal (to the surface) force acting on this element,

dF_s = (S1/R1 + S2/R2)*dAdt

where R1, R2 are the main surface radii of curvature (at the point of evaluation) and S1, S2 are stresses along the directions of the corresponding axes of curvature.

The ball rotates with an angular velocity w. Work in the rotating reference frame. The surface element is acted upon by two forces. The elastic force simplifies to

dF_s = (S1 + S2)*dAdt/R

since R1 = R2 = R, the radius of the sphere, pulling the surface element inwards towards the rotation axis. The second force is centrifugal

dF_c = R*w^2*dm

dm is mass of the surface element, given by

dm = dAdt*rho

rho is density. Then,

dF_c = R*w^2*rho*dAdt

acting outward from the rotation axis. The surface element remains stationary in the rotating frame (until the sphere unbinds), the two forces dF_s and dF_c are equal.

R*w^2*rho*dAdt = (S1 + S2)*dAdt/R

Cancel common factors, then

(R*w)^2 = (S1 + S2)/rho

R*w is v, the velocity of the equatorial point. Neither S1 nor S2 can be larger than tensile strength S. Then,

v^2 <= 2*S/rho

with = obtaining at the limit of material unbinding. Radii cancel.

S/rho, energy/mass, is binding energy/unit mass of material. The ball remains bound if its kinetic energy/unit mass (in any locality) is less than binding energy/unit mass.

Like Wayne says, the ‘centripetal’ force is the inward force of the atomic bonds causing the carbon atoms to follow the curved path around the center of rotation. The force is real, and causes the acceleration (F=ma) of the carbon atoms so that they follow the curved path around the center of rotation.

The outward force is centrifugal, and is an illusionary force. It is not really there (not causing a real acceleration via F=ma) but arises due to the rotating reference frame if you have the point of view of ‘following’ one of the carbon atoms as the graphene rotates.

“Let me clear and blunt here: that’s wrong. Centrifugal force is every bit as real as centripetal force. It’s just in a different frame. “Centripetal” means “center-seeking”, and “centrifugal” means “outwards-seeking” or, more literally, “center-fleeing”. You’d think these are opposites, but they are in fact the same thing! It just depends on your point of view.”

Well, to be technical, they are not really the same thing, but just equal but opposite forces , in the same sense that gravity force, and normal (support) forces are equal but opposite for a person at rest on earth’s surface.

Let me illustrate: suppose you are in a turning car and are pushed up against the side door. In your frame of reference, you are under compression, but at rest. And yes, there are two equal but opposite forces: the “centrifugal force” pulling outward (really a gravitational force), and a “centripetal force” pushes inward , which is created by the compressional forces of the door onto you.

The thought that I have is to move this to the real world, the sheets will have to be much thicker than one atom. Once the layered sheet equations are possible, I wonder on the strength of the material. Will cross links in the “vertical” plane be as strong as the flat sheets. I hope so because as others have wanted, Space Elevators… Tethered rotational spacecraft pairs… etc. I wonder of the electrical properties of this stuff. Higher temp superconductors.

If these sheets can be made in significant sizes (say, 100 cm in radius) and multilayered, they might be ideal as an energy storage device. I don’t have the equations to calculate the energy storage(dependent upon radial velocity and mass, which should be an integral equation) but from what I recall, the best inertial storage systems are about as good as a Lithium/ion battery. These carbon sheets might be able to improve that by a couple of orders of magnitude.

Imagine such a storage device for your car, 1000 miles on a “fill up” and only .2 meters in diameter.

Gentle suggestion (12): No, I was just rounding up the radius to make it easier to show the math. If the flakes are square there would be factors of sin(45) in there and all sorts of things, so I just made the radius 1μ to make it easy.

@2: No nanodiamonds have the same strength characteristics as regular diamonds. Graphene’s strength comes the fact it’s a single sheet and carbon-carbon bonds bonds in graphene are much stronger than in diamonds (sp2 vs. sp3 in chemist speak) If I recall correctly, the latest test showed graphene to be approximately 117 times stronger than steel. Granted that was on a piece of graphene 1.5 micrometers across.

@14: That’s a good question, no one knows right now because it is very difficult to produce graphene sheets over a 100 micrometers square. However, if anyone finds out how to make a 5 x 5 cm sheet, they’ll end up being the richest person in the world.

“Space elevators would be the ideal solution to the problem of getting material (& people) ”

Ideal for fragile materials, not so much for people, since it would take about two weeks to get to GeoSync orbit. Two weeks in an elevator is not my idea of space travel AND the ribbon is subject to erosion from space debris(which would, of course, require constant repair).

Still, it’s many magnitudes of improvement in energy efficiency over chemical rockets.

“Even some high-mass main sequence stars spin astoundingly fast. So much so that some – three notable examples being Altair, Regulus and, most of all, Achernar – are distorted from spherical globes to elliptical egg-shapes.”

Um, not quite. An egg is prolate (its symmetry axis is its long axis), while a spinning object in hydrostatic equilibrium (like a lone, spinning star) is oblate (its symmetry axis is its short axis). Think of prolate as a sphere that’s been stretched along one axis, while prolate is a sphere that’s been squashed along one axis.

When I hear about things spinning this fast I can’t imagine how that is possible. How can an object complete 1 million 360 degree rotations in “One Mississippi.” It just seems like there is not enough time to do 1 million of anything during that period. That means in 1/10 of a second it has already spun 100,000 complete revolutions?

Chief @ 28: Multi-layered graphene is very common and people used to use them every day (less so nowadays, I guess). They are called pencil “leads”. The weak bonds between the sheets makes them so ideal for writing and drawing with. If you want strong bonds in all directions you’d better look at diamonds.
Cheers, Regner

Wow, I had thought about doing the same thing when I heard this news story pop up on Slashdot. Then I got sidetracked looking up the formulas for time dilation instead in response to a story I was thinking writing…

someone estimated the G’s acting on the astronauts in Jules Verne’s From the Earth to the Moon when the canon launched them, at 2500 G’s. a force that would render the organic passengers (I believe there was a dog along as well) a 2 centimeter goo on the aft bulkhead.

Chief @ 45: It is not impurities that makes graphene layers weakly bonded. It is an intrinsic property of graphene. The only way to change that would be to dope it in some way – adding impurities – but I don’t know if you can actually construct a much tighter inter-layer bond in that way.
Cheers, Regner

Graphene is actually quite a bit stronger than diamond. Diamond is *harder*, not *stronger*. The two things are not the same.

@Nicholas: “When I hear about things spinning this fast I can’t imagine how that is possible.”

If you *really* want your mind blown, one theory of the origin of ‘spin’ for electrons suggests they ‘jitter’ in very small circular motion around 1.6 * 10 ^21 times per second. Look up ‘Zitterbewegung’.

@Gary Ansorge: “If Reverend J has the right differential(117 times as strong as steel) this would imply an energy storage density of about 301,000 W-sec/kg.”

Just don’t get in an accident. The problem with very high energy density kinetic storage is that it tends to go ‘boom’ quite violently when mis-handled.

@Tibs: “Well millisecond pulsars “blip” once every millisecond or so: 1/1000th of a second. These carbon flakes are spinning a thousand times as fast as THAT.”

The flakes are spinning a 1000 times as fast as a millisecond pulsar, but the pulsar has something like five billion times the radius. The implied centripetal acceleration is therefore about 5000 times as large (a = r*w^2, ignoring relativistic effects) or (*very* approximately) a couple of hundred billion Gs. Of course, there is something like 400 billion Gs typically holding the neutron star together….

[space elevators are] Ideal for fragile materials, not so much for people, since it would take about two weeks to get to GeoSync orbit. Two weeks in an elevator is not my idea of space travel AND the ribbon is subject to erosion from space debris(which would, of course, require constant repair).

Pardon my ignorance here but why would it have to take two weeks? Couldn’t they make the “lift” rise faster?

@Messier Tidy Upper: I don’t know how long it would take, but a space elevator would (I think?) require its upper end to be geosynchronous, as Gary Ansorge said. For the Earth, geosync orbit is over 20,000 miles up into space — almost a tenth of the way to the moon!

There are satellites that high, but they fly freely on a rocket until they’re in place. I imagine that climbing a space elevator in a rocket kinda defeats the purpose of the thing, so you’d probably go a lot slower.

@36. Gary Ansorge Says: “Two weeks in an elevator is not my idea of space travel AND the ribbon is subject to erosion from space debris(which would, of course, require constant repair).”

Two weeks? That’s the speed of a normal Earth-bound elevator (about 65 MPH). I figure if we have the technology to build the space elevator, we’ll have techniques to go much faster than that, such as magnetic suspension so there’s no physical contact with the column. Start out at 100 MPH or so for an hour to clear the atmosphere, then crank it up to 1,000. You’ll be in geosync in less than a day.

Even if it took longer, do you think these would be like your run-of-the-mill office elevator car? I envision them as being airliner size, only arranged differently. You can get up, walk around, watch a movie and chat with friends about what you’re going to do when you get there.

Clarke wrote a whole novel about it (“Fountains of Paradise”) and even incorporated them into “3001: The Final Odyssey. If you don’t like the zero-g of geosync, you can get out at any gravity level you want on the way up. He had large recreational areas at 1/10 g where human powered flight is not only possible, but recreational.

“Would that be including millisecond pulsars / magnetars and black holes or not?”

The fastest rotation of a solar-mass black hole is 62 microseconds per revolution. That’s only about 16,000 revolutions per second, a long way short of graphene. Remember, the circumference can’t go faster than the speed of light. A solar-mass black hole has a circumference of 18.5 km. Any faster rotation and the speed at the surface would exceed 300,000 km/s, the speed of light.

Larger black holes will rotate even slower. A primordial black hole could go much faster, but none of those have been found – yet. If they do not exist, then the smallest possible black hole is is about 1 sol (after mass ejection in the supernova explosion)

The acceleration in this case is not so important. What is the angular momentum? Without knowing of any preferred orientation you can assume a rotation axis with maximum momentum and one with minimum momentum and an equitable distribution of every state in between. Despite the high claimed rotational rate, the flakes weigh virtually nothing and so there is hardly any energy involved.

The fastest rotation of a solar-mass black hole is 62 microseconds per revolution. That’s only about 16,000 revolutions per second, a long way short of graphene. Remember, the circumference can’t go faster than the speed of light. A solar-mass black hole has a circumference of 18.5 km. Any faster rotation and the speed at the surface would exceed 300,000 km/s, the speed of light.

Thanks for that info. & explanation – much appreciated.

A primordial black hole could go much faster, but none of those have been found – yet. If they do not exist, then the smallest possible black hole is is about 1 sol (after mass ejection in the supernova explosion.)

Only one solar mass? I would’ve thought you couldn’t get any Black Holes less massive than 3 solar mass or so or you’d end up with a neutron star being produced instead? In fact under 1.4 solar mass and you’ve got a white dwarf not a neutron star or so I, perhaps wrongly, understand things, yes?

I suppose there’s evapouration via Hawking radiation to take into account which could reduce the original mass down to one solar but won’t that take many aeons and has there actually been sufficent time in the cosmos so far for that to have ocurred yet?

I recall reading about primordial black holes created in or just after the Big Bang but they seem to have gone out of vogue and I haven’t heard anything about them in years. Has there been any developments with primordial Black Hole theory disproving the idea or is it just lack of observational evidence?

Andre Geim and Konstantin Novoselov, at my alma mater and current place of employment (University of Manchester, UK) have been awarded the Nobel Prize in physics for
” groundbreaking experiments regarding the two-dimensional material graphene”

you did it exactly correct. Why do you think you did it wrong? remember, if you are on a rotating platform , an object like (2) on that platform will have an “outward” acceleration for a instant (instantaneous acceleration). I believe that the math works out exactly that way.

I know it is hard to picture “centripetal” or centrifugal accelerations. For example, it does not seem that an object orbiting the earth is “accelerating” at all, but it is.

Nobel Prize for Graphene!!!!!!!!!!!!!!!!!!!!!
“Andre Geim (pictured) and Konstantin Novoselov win the 2010 Nobel Prize in Physics for their experiments on graphene.” (Wiki)
Atk
Ops: “Geim shared the 2000 Ig Nobel Prize with Sir Michael Berry of Bristol University, for levitating the frog. His award of the Nobel Prize for Physics in 2010 made him the first person to win an Ig followed by the real version.” (Wiki)

It’s because of the hybridization of the orbitals in the carbon which give it’s the bend. The carbon is called “sp2” which all it’s electron orbitals are in a plane but they repel each other so they try and get as far part as possible, thus they are 120 degrees from each other.

The explanation I got that made sense (as much as these things ever do) is that the apparent mass of an object increases as it approaches the speed of light. At the speed of light the apparent mass goes to infinity and all the energy in the universe cannot increase the speed up the object any further.

A black hole will have a finite amount of energy associated with it’s spin. It will be a prodigious amount of energy mind you, but never infinite. That’s what caps it’s spin rate.

Yes, and not just that. Buckytubes are cylinders of graphene. The only thing that makes them different is that the edges are curled around and intersect the same plane of material. The result is an enormously strong shape with many neato properties.

Reading some of the science fiction Golden Age authors like Campbell I remember spaceships built from diamond, formed an atom at a time. They knew what was coming. They didn’t have the vocabulary for it, but they knew.

Our progress with materials sciences has been so amazing. It frustrates me that we really haven’t had an equivalent breakthrough in energy generation, propulsion, or energy storage. Sure there have been minor increases in efficiencies, but we still use lead acid batteries to power most electric cars. Even our best NiCds and NiMhs don’t have much better energy storing capacity.

C’mon Phil, when are the boys going to give us antigrav cars, and powercells for our handheld devices that can power them for years? I want my jetpack, damnit.

You are right in saying that the star that produces a blck hole has to be about 3 solar masses. I’m not sure, however, what is the minimum left after the imploding star has thrown off it’s outer shell as a supernova. A quick google search revealed that the smallest known black hole is 3.8 sols ( http://www.space.com/scienceastronomy/080401-smallest-blackhole.html ).

Anyway, I did say “about” 1 solar mass, so I’m certainly within an order of magnitude. And, you’d need to go a lot smaller to get to 1,000,000 rps. As the circumference of the hole is proportional to its mass, down to 1/16th solar mass in fact, way too small to get it from a supernova

As for primordial black holes, none have been found, and they are certainly a lot rarer than theorists believed.

“,,,and powercells for our handheld devices that can power them for years”

See: Nuclear lightbulb.

“With the nuclear lightbulb, charged particles from the nuclear reaction (usually tritium) excite a gas such as argon or xenon, both of which emit light in the ultraviolet range. Sodium, potassium, or mercury, which emit light in the blue to green range, also can be used. The ultraviolet or bluegreen radiation then activates the photovoltaic cell to produce energy.”

Budda:
” It frustrates me that we really haven’t had an equivalent breakthrough in energy generation, propulsion, or energy storage. Sure there have been minor increases in efficiencies, but we still use lead acid batteries to power most electric cars. Even our best NiCds and NiMhs don’t have much better energy storing capacity.”

my two cents, energy that is usable by society is a lot more elusive than we think. Has something to do with entropy, energy is the ultimate low entropy stuff and tough to get at

Graphene would make a great substrate for a gargantuan space telescope mirror- I’ve written about this here and elsewhere before. If made right, its completely smooth as far as photons are concerned (though there are ripples and waves that might need to be suppressed or at least accounted for in the electronic management of image cumulation and analysis). One could make a mirror miles, or millions of miles across, and use light pressure from the sun to keep it parabolic (or whatever shape you want, hyperbolic, etc.). The reflective coating is a bigger supply issue, though- where will you get all that metal or other materials? Nickel might be ideal- plenty of that in asteroids, and its awfully reflective. In space it shouldn’t oxidize too quickly. Spinning up a giant graphene mirror might help keep it tense, so it would be good to know what kind of forces it could take before breaking up.